GOZUNTA
Try it now
All courses Geometry · G-GMD.1 22 of 57
Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.

Use Cavalieri's principle to compare volumes

Problem
Using \(\text{Cavalieri}'s~\text{principle}\), what can you conclude about the volumes of \(\text{two}~\text{prisms}\) with equal height and equal cross-sectional area at every level?
Geometric source configuration showing only the supplied dimensions and construction. Open full size
Your answer
Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
With a free account

Read the steps or watch the work.

Every problem included with your free account keeps its hints, complete written solution, and step-by-step learning video available for review.

Create a free account
With paid access

Open the complete course.

A subscription unlocks every problem type and variant in your selected course, not only the starter set.

Compare plans

Hint

Identify the two facts given: the prisms have the same height, and at every level their cross-sectional areas are equal.

Cavalieri's principle says that if two solids have equal heights and equal cross-sectional area at every height, then their volumes are equal.

Solution walkthrough

01

List Cavalieri's two hypotheses

\[\text{same}~\text{height}~h;~\text{equal}~\text{cross}-\text{sectional}~\text{area}~\text{at}~\text{every}~\text{level}\]

Both facts come directly from the prompt. The inspected comparison graphic reinforces that the solids extend through the same height and have matching horizontal slice areas at corresponding levels.

02

State why the principle applies

\[\text{same}~\text{height}~+~\text{equal}~\text{area}~\text{of}~\text{every}~\text{corresponding}~\text{slice}~->~\text{equal}~\text{volume}\]

Cavalieri's principle compares the accumulated volume layer by layer. It does not require congruent solids or the same slant.

03

Apply the implication

\[V_1=V_2\]

Since every corresponding cross-section has equal area over the same total height, adding all of the equal-volume thin layers gives equal total volumes.

04

Interpret the result

\[\text{the}~\text{two}~\text{prisms}~\text{have}~\text{the}~\text{same}~\text{volume}\]

No numerical calculation is needed; the supplied slice and height conditions determine the comparison.

+

Another way

  1. Imagine approximating each solid by many thin layers: equal layer area times equal layer thickness gives matching partial volumes, and their totals match.

!

Common mistake

Do not require the prisms to have the same overall shape. Cavalieri's principle permits different shapes or slants as long as the heights and every corresponding cross-sectional area match.

Completed geometric argument with matched measures, limiting relation, and formula conclusion.
equal volumes

Solution walkthrough video